2.1 Equations

Using the Einstein summation convention the equations describing the motion of a relativistic fluid are given by the five conservation laws,
μ (ρu );μ = 0, (1 ) T μν;ν = 0, (2 )
where μ,ν = 0,...,3, and where ;μ denotes the covariant derivative with respect to coordinate μ x. Furthermore, ρ is the proper rest mass density of the fluid, μ u its four-velocity, and μν T is the stress-energy tensor, which for a perfect fluid can be written as
μν μ ν μν T = ρhu u + pg . (3 )
Here, gμν is the metric tensor, p the fluid pressure, and h the specific enthalpy of the fluid defined by
p- h = 1 + 𝜀 + ρ, (4 )
where 𝜀 is the specific internal energy. Note that we use natural units (i.e., the speed of light c = 1) throughout this review.

In Minkowski spacetime and Cartesian coordinates (t,x1,x2,x3), the conservation equations (1View Equation, 2View Equation) can be written in vector form as

∂u ∂Fi (u ) --- + ----i-- = 0, (5 ) ∂t ∂x
where i = 1 ,2 ,3. The state vector u is defined by
1 2 3 T u = (D, S ,S ,S ,τ) (6 )
and the flux vectors Fi are given by
Fi = (Dvi,S1vi + pδ1i,S2vi + pδ2i,S3vi + pδ3i,Si − Dvi )T. (7 )
The five conserved quantities D, S1, S2, S3, and τ are the rest mass density, the three components of the momentum density, and the energy density (measured relative to the rest mass energy density), respectively. They are all measured in the laboratory frame, and are related to quantities in the local rest frame of the fluid (primitive variables) through
D = ρW, (8 ) Si = ρhW 2vi, i = 1, 2,3, (9 ) 2 τ = ρhW − p − D, (10 )
where vi are the components of the three-velocity of the fluid
ui vi = -0, (11 ) u
and W is the Lorentz factor,
W = u0 = √---1-----. (12 ) 1 − vivi
The system of Equations (5View Equation) with Definitions (6View Equation, 7View Equation, 8View Equation, 9View Equation, 10View Equation, 11View Equation, 12View Equation) is closed by means of an equation of state (EOS), which we shall assume to be given in the form
p = p(ρ,𝜀). (13 )

In the non-relativistic limit (i.e., v ≪ 1, h → 1) D, Si, and τ approach their Newtonian counterparts ρ, ρvi, and ρE = ρ𝜀 + ρv2∕2, and Equations (5View Equation) reduce to the classical ones. In the relativistic case the equations of system (5View Equation) are strongly coupled via the Lorentz factor and the specific enthalpy, which gives rise to numerical complications (see Section 2.3).

In classical numerical hydrodynamics it is very easy to obtain i v from the conserved quantities (i.e., ρ and ρvi). In the relativistic case, however, the task to recover (ρ,vi,p) from (D, Si,τ) is much more complicated. Moreover, as state-of-the-art SRHD codes are based on conservative schemes where the conserved quantities are advanced in time, it is necessary to compute the primitive variables from the conserved ones one (or even several) times per numerical cell and time step making this procedure a crucial ingredient of any algorithm (see Section 9.2).

  Go to previous page Go up Go to next page